Geodesic distances in Liouville quantum gravity

Abstract

In order to study the quantum geometry of random surfaces in Liouville gravity, we propose a definition of geodesic distance associated to a Gaussian free field on a regular lattice. This geodesic distance is used to numerically determine the Hausdorff dimension associated to shortest cycles of 2d quantum gravity on the torus coupled to conformal matter fields, showing agreement with a conjectured formula by Y. Watabiki. Finally, the numerical tools are put to test by quantitatively comparing the distribution of lengths of shortest cycles to the corresponding distribution in large random triangulations.

21 pages, 8 figures

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Cited by in corpus (10)

• The fractal dimension of Liouville quantum gravity: universality, monotonicity, and bounds
• Liouville quantum gravity with matter central charge in $(1,25)$: a probabilistic approach
• Upper bounds on Liouville first passage percolation and Watabiki's prediction
• Scale-dependent homogeneity measures for causal dynamical triangulations
• The dimension of the boundary of a Liouville quantum gravity metric ball
• Nesting statistics in the O(n) loop model on random planar maps
• Precision measurements of Hausdorff dimensions in two-dimensional quantum gravity
• Non-universality for first passage percolation on the exponential of log-correlated Gaussian fields
• Liouville first passage percolation: the weight exponent is strictly less than 1 at high temperatures
• Regular Polygon Surfaces